The Korteweg–de Vries equation \[ u_t + uu_x + u_{xxx} = 0\] is a nonlinear partial differential equation arising in the study of a number of different physical systems, e.g., water waves, plasma physics, anharmonic lattices, and elastic rods. It describes the long time evolution of small-but-finite amplitude dispersive waves. From detailed studies of properties of the equation and its solutions, the concept of solitons was introduced and the method for exact solution of the initial-value problem using inverse scattering theory was developed. A survey of these and other results for the Korteweg–deVries equation are given, including conservation laws, an alternate method for exact solution, soliton solutions, asymptotic behavior of solutions, Backlund transformation, and a nonlinear WKB method. The recent literature contains many extensions of these ideas to a number of other nonlinear evolution equations of physical interest and to other classes of equations. Some of these equations and results are indica...

Decay of Solutions of Systems of Nonlinear Hyperbolic Conservation Laws

The Korteweg–deVries Equation: A Survey of Results

Measure-valued solutions to conservation laws

Publisher Summary This chapter provides an overview of shock waves and entropy. It describes systems of the first order partial differential equations in conservation form: ∂ t U + ∂ X F = 0, F = F(u). In many cases, all smooth solutions of the first order partial differential equations in conservation form satisfy an additional conservation law where U is a convex function of u. The chapter discusses that for all weak solutions of ∂ t u j +∂ x f j = 0, j=1,…, m, f j =f j (u 1 ,…, u m ), which are limits of solutions of modifications ∂ t u j +∂ x f j = 0, j=1,…, m, f j =f j (u 1 ,…, u m ) , by the introduction of various kinds of dissipation, satisfy the entropy inequality, that is, ∂ t U + ∂ x F≦ 0. The chapter also explains that for weak solutions, which contain discontinuities of moderate strength, ∂ t U + ∂ x F≦ 0 is equivalent to the usual shock condition involving the number of characteristics impinging on the shock. The chapter also describes all possible entropy conditions of ∂ t U + ∂ x F≦ 0 that can be associated to a given hyperbolic system of two conservation laws.

Shock Waves and Entropy

A convergence theorem for the method of artificial viscosity applied to the isentropic equations of gas dynamics is established. Convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves.

methods

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Convergence of the Viscosity Method for Isentropic Gas Dynamics

Systems of conservation laws arise naturally in physics and chemistry. To understand them and their consequences (shock waves, finite velocity wave propagation) properly in mathematical terms requires, however, knowledge of a broad range of topics. This book sets up the foundations of the modern theory of conservation laws, describing the physical models and mathematical methods, leading to the Glimm scheme. Building on this the author then takes the reader to the current state of knowledge in the subject.The maximum principle is considered from the viewpoint of numerical schemes and also in terms of viscous approximation. Small waves are studied using geometrical optics methods. Finally, the initial-boundary problem is considered in depth.Throughout, the presentation is reasonably self-contained, with large numbers of exercises and full discussion of all the ideas. This will make it ideal as a text for graduate courses in the area of partial differential equations.

Decay of Solutions of Systems of Nonlinear Hyperbolic Conservation Laws

Citations

The Korteweg–deVries Equation: A Survey of Results

Measure-valued solutions to conservation laws

Shock Waves and Entropy

Convergence of the Viscosity Method for Isentropic Gas Dynamics

Systems of Conservation Laws 1: Hyperbolicity, Entropies, Shock Waves

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